Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,18)(13,22)(14,20)(15,17)(19,23)(21,24)(25,26,30,34)(27,29,33,31)(28,36,37,32)(35,38,40,39);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,15)(13,21)(14,19)(17,18)(20,23)(22,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,17)(11,12)(19,22)(20,21)(28,37)(32,36)(35,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,19,24,22)(25,28,30,37)(26,32,34,36)(27,35,33,40)(29,39,31,38);(9,13,15,23)(10,19,17,22)(11,21,12,20)(14,18,24,16)(25,27)(26,29)(28,35)(30,33)(31,34)(32,39)(36,38)(37,40);(9,14,15,24)(10,20,17,21)(11,19,12,22)(13,16,23,18)(25,29)(26,33)(27,34)(28,38)(30,31)(32,35)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8625564);